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# [Phys-L] Fwd: relativistic acceleration of an extended object

• From: John Mallinckrodt <ajm@cpp.edu>
• Date: Sun, 9 Nov 2014 15:43:21 -0800

Yesterday Moses Fayngold defended his earlier statement:

3). Due to relativity of time, the object's clocks along N read different times at one moment of A-time.

Again, no. You are misapplying a result of relativity that applies only to an array of comoving clocks that--importantly--are synchronized in their own rest frame. Let's review:

I. ASSUMPTIONS

We have a linear array of clocks, perhaps (but irrelevantly) attached to points along a deformable object. All clocks are initially at rest in inertial frame A. All clocks are initially synchronized with themselves and other clocks that will remain stationary in A.

The clocks in the linear array begin accelerating in a direction along the line of clocks so that there are identifiable "front" and "rear" clocks, F and R respectively. The accelerations begin simultaneously in A and the acceleration versus time profile is identical for each of the accelerating clocks as measured in A.

The accelerations end when each clock reaches speed V as measured in A.

II. THINGS THAT FOLLOW IMMEDIATELY FROM THESE ASSUMPTIONS

1. The proper (i.e. "felt") acceleration versus proper time (i.e. the reading of the clock itself) profile is also identical for each of the accelerating clocks.

2. At any given instant in frame A each accelerating clock is moving at the same speed and displays the same time reading.

Corollaries: In reference frame A the accelerating clocks maintain their initial spacing, all reach speed V simultaneously, and all display the same time when they reach speed V.

Note: The accelerating clocks all run slow as determined by observers in A and, therefore, read earlier times than clocks in A as measured in A.

3. When all of the clocks have stopped accelerating they will be found to form a linear array that is at rest in a new inertial frame, B, that moves at speed V relative to A.

4. The final spacings of the clocks in B will be larger than the respective initial spacings in A by the standard relativistic factor, gamma(V).

5. In frame B after the acceleration phase has ended, the array of now motionless clocks is not synchronized. In particular, clock F will read a later time than clock R. Indeed, the clocks will show precisely the times required by the fact that they remain, at all times during and after the acceleration phase, synchronized as observed in A.

Note: This result is nicely compatible with the qualitative requirement of General Relativity that "higher clocks run faster than lower clocks." During the acceleration phase, the clocks experience the effects of a gravitational field in which F is "higher" than R.

John Mallinckrodt
Cal Poly Pomona
johnmallinckrodt.com